Second Derivative Calculator
The second derivative reveals the concavity of a function: whether a curve bends upward or downward and where its inflection points lie. This calculator estimates both the first and second derivative at a point using the central finite-difference formulas, which need only three function values: one step before the point, at the point, and one step after. You supply those three values and the step size h. The tool returns the slope from the central difference and the concavity from the central second difference, both standard, source-able numerical methods.
Finite-difference formulas
f'(x) = (f(x+h) - f(x-h)) / (2h)
f''(x) = (f(x-h) - 2*f(x) + f(x+h)) / h^2
Concave up if f''(x) > 0
Concave down if f''(x) < 0
The central difference for the first derivative averages the forward and backward slopes. The central second difference compares the two outer values against twice the central value, scaled by the step size squared.
Notes on numerical derivatives
- The three sample points must be evenly spaced by the step size h.
- A smaller h improves accuracy until floating-point rounding error takes over.
- The second derivative is exact for any quadratic regardless of step size.
- A positive second derivative indicates a curve that is concave up at the point.
- A negative second derivative indicates a curve that is concave down at the point.
Second derivative: frequently asked questions
What is the second derivative?
The second derivative measures the rate of change of the first derivative, in other words how the slope of a curve is itself changing. A positive second derivative means the curve is concave up; a negative second derivative means it is concave down.
How does this calculator estimate the second derivative?
It uses the central finite-difference formula. Given the function values at three evenly spaced points (one step h before, at, and after the point), the second derivative is approximately (f(x-h) - 2*f(x) + f(x+h)) divided by h squared.
How accurate is the central difference formula?
The central second-difference formula has an error proportional to h squared, so a smaller step gives a more accurate estimate, until rounding error from subtracting nearly equal numbers begins to dominate at very small h.
What does the first derivative output tell me?
The first derivative is estimated with the central difference (f(x+h) - f(x-h)) divided by 2h. It gives the instantaneous slope of the function at the point, indicating whether the function is rising or falling there.
What does a zero second derivative mean?
A point where the second derivative is zero and changes sign is an inflection point, where the curve switches between concave up and concave down. A zero second derivative alone does not guarantee an inflection; the sign must change.
Official sources
- NIST Digital Library of Mathematical Functions: Differentiation.
- NIST Digital Library of Mathematical Functions: Calculus of one variable.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.